Arranging books in bookshelves with the capacity of each shelf given
There are k identical bookshelves in which each shelf cannot contain m or more books. In how many ways can n distinct books be arranged on these k bookshelves?
If there is no condition on the capacity of each shelf, the number of ways to arrange books equals , where L(n,i) denotes the Lah number. However, because of that constraint, I have trouble in solving the problem.
I tried several ways to solve this problem by separating the cases via (1) the number of shelves which contains the full-number of books, or (2) the number of non-empty shelves. For the second trial, I observed that, if j denotes the number of non-empty shelves, then the number of ways to arrange the books is zero if .
However, these methods does not proceed quite well, since it looks like these methods result in the recurrence relation rather than the exact form of the number.
For the related concepts, I have studied Catalan number, (both signed and unsigned) first and second Stirling number, Bell and Lah number, and the integer partition.
Any insight or comment are welcomed.